Preheat oven to 425ºF. Line a large, rimmed baking sheet with foil. Mist a 3-quart shallow microwave- and oven-safe baking dish with cooking spray. Combine sage and thyme in a small bowl. In baking dish, layer 1/3 of potatoes and sweet potatoes, alternating the slices. Sprinkle with 1/3 of herb mixture and 1/3 each salt and pepper. Repeat layering 3 times.
Combine half-and-half and broth and pour over potatoes. Place dish on baking sheet and cover with foil. Bake 50 minutes. Uncover, sprinkle with cheese and bake until golden and bubbly, 20 to 25 minutes longer. Let rest for 10 minutes before serving.